Learning Partially Observed PDE Dynamics with Neural Networks

Ibrahim Ayed, Emmanuel De Bézenac, Arthur Pajot, Patrick Gallinari

Abstract: Spatio-Temporal processes bear a central importance in many applied scientific fields. Generally, differential equations are used to describe these processes. In this work, we address the problem of learning spatio-temporal dynamics with neural networks when only partial information on the system's state is available. Taking inspiration from the dynamical system approach, we outline a general framework in which complex dynamics generated by families of differential equations can be learned in a principled way. Two models are derived from this framework. We demonstrate how they can be applied in practice by considering the problem of forecasting fluid flows. We show how the underlying equations fit into our formalism and evaluate our method by comparing with standard baselines.

Keywords: deep learning, spatio-temporal dynamics, physical processes, differential equations, dynamical systems